Why would neural networks be a particularly good framework for "embodied AI"?
To my mind the essential reason why neural networks and the brain are powerful is that they create a hierarchical model of data or of the world. If you ask why that makes them powerful, well, that's just the structure of the world. If you are stalked by a wolf, it's not like its upper jaw will attack you frontally, while his lower jaw will attack you from behind. If you want to respond to the threat with a feasible computational effort, you'll have to treat the wolf as one entity. Providing these kinds of entities or concepts from the raw bits and bytes of input is what a hierarchical representation does. 

Now, this is quite intuitive for sensory information: lashes, iris, eyebrow make up an eye, eyes, nose and mouth make up a face and so on. What is less obvious, is the fact that motor control works exactly the same way! Only in reverse. If you want to lift your arm, you'll just lift it. But for your brain to actually realise this move, the high level command has to be broken down into precise signals for every muscle involved. And this is done by propagating the command down the hierarchy. 

In the brain these two functions are strongly intertwined. You use constant sensory feedback to adapt your motor control and in many cases you'd be incapable of integrating your stream of sensory data into a coherent representation if you didn't have the additional information of what your body is doing to change that stream of data. Saccades are a good example for that. 

Of course this doesn't mean that our cognitive functions are dependent on the processing of sensorimotor information. I would be surprised if a pure thinking machine wouldn't be possible. There is however a specific version of this "embodied intelligence hypothesis" that sounds plausible to me: 

Creating high level cognitive concepts with unsupervised learning is a really difficult problem. Creating high level motor representation might be significantly easier. The reason is that there is more immediate useful feedback. I have been thinking about how to provide a scaffolding for the learning of a hierarchy of cognitive concepts and one thing I could imagine is that high level cognitive concepts basically hitch a ride with the motor concepts. Just think of what a pantomime can express with movement alone.